The set \(\mathcal {D}\)  is clearly finite, and its cardinality is bounded as \(P := |\mathcal {D}| \le 2r\left(\frac{e(n-1)}{r}\right)^r\) , where \(r := \mathrm {rank}({\mathbf {X}})\)  [1]}, [2]}. Through convex duality analysis, we can express an equivalent convex optimization problem by enumerating over the finite set of arrangements \(\lbrace \mathbf {D}_j\rbrace _{j=1}^P\) . We define the following norm
\(&\Vert \mathbf {Z}\Vert _{*, \mathrm {K}} := \min _{t \ge 0} t \; \text{s.t.}\, \mathbf {Z} \in t\mathcal {C}\\&\mathcal {C} := \mathrm {conv}\lbrace \mathbf {Z} = \mathbf {u}\mathbf {v}^\top : \; \mathbf {K}\mathbf {u} \ge 0, \, \Vert \mathbf {Z}\Vert _* \le 1, \mathbf {v} \in \mathbb {R}^c\rbrace . \nonumber \) 
